Geometric approaches to Lagrangian averaging

TL;DR

This paper reformulates Lagrangian averaging theories like GLM using a geometric, coordinate-free approach, clarifying their structure and deriving governing equations for mean flow and wave activity in stratified flows.

Contribution

It introduces a geometric, coordinate-free formulation of Lagrangian averaging theories, unifying different approaches and deriving new governing equations for mean flow and wave interactions.

Findings

Reformulation of Lagrangian averaging in geometric terms

Derivation of governing equations for mean flow and wave activity

Unification of GLM and glm formulations through geometry

Abstract

Lagrangian averaging theories, most notably the Generalised Lagrangian Mean (GLM) theory of Andrews & McIntyre (1978), have been primarily developed in Euclidean space and Cartesian coordinates. We re-interpret these theories using a geometric, coordinate-free formulation. This gives central roles to the flow map, its decomposition into mean and perturbation maps, and the momentum 1-form dual to the velocity vector. In this interpretation, the Lagrangian mean of any tensorial quantity is obtained by averaging its pull back to the mean configuration. Crucially, the mean velocity is not a Lagrangian mean in this sense. It can be defined in a variety of ways, leading to alternative Lagrangian mean formulations that include GLM and Soward & Roberts' (2010) glm. These formulations share key features which the geometric approach uncovers. We derive governing equations both for the mean flow and for wave activities constraining the dynamics of the pertubations. The presentation focusses on the Boussinesq model for inviscid rotating stratified flows and reviews the necessary tools of differential geometry.